Numbers: Divisibility Rule (21-30)

In many competitive questions, it becomes very important to find whether given number is divisible by another given number or not. It is required when we solve HCF (Highest Common Factor), LCM (Least Common Multiple), Composite numbers, Co-Prime numbers, multiples, factors, etc.

So this topic is extensively important in understanding many parts of number mathematics and algebra as well because all operations in number mathematics are always true for algebra and other mathematics also.

Divisibility Rule (Divisibility Test - 21 to 30)

Divisibility test means testing of a number whether it is divisible completely by another given number. It is always means that the first number (Dividend) and the second number (Divisor) are integers. Result of a division is called Quotient and if something is left, it is called a remainder.

For Example:

456382 ÷ 12 = (Q=38031 and R=10)

Important:

"If in a division, quotient is an integer, and remainder is zero, it means that dividend is completely divisible by divisor."

"If in a division, either quotient is a fraction with remainder zero or quotient is integer with remainder non-zero, it means that dividend is not completely divisible by divisor."

21. Divisible By 21: If a number is divisible by 3 as well as divisible by 7, then the number is completely divisible by 21.

So it is very clear that the number should follow both the rules of divisibility test by 3 and divisibility test by 7 simultaneously.

Note: If number is not divisible by any one of 3 and 7, then number will not be divisible by 21. If number is divisible by 3 by not divisible by 7, it is not divisible by 21. Similarly, if number is not divisible by 3 but divisible by 7, it is not divisible by 21.

Step for Test: 1. Divisibility Test By 3:      Sum all the digits     If final sum is divisible by 3, then the given number is divisible by 3. Then Proceed further.     If final sum is not divisible by 3, then the given number is not divisible by 3. Then number is not divisible by 7 2. Divisibility Test by 7:     Write the last digit of number i.e. one's place of the number.     Double it (Means multiply it by 2)     Subtract this number from the remaining portion of the number. We will obtain a result.     Repeat above three steps for the result obtained many times until we get a number less than 70.     If final result is divisible by 7, then the given number is divisible by 7. 3. If number is divisible by 3 as well as 7 then number is divisible by 21

For Example:

Let us check whether 98994 is divisible by 21. 

Step 1: Test of divisibility by 3

     Sum all digits           = 9 + 8 + 9 + 9 + 4 = 39

     Again sum all digits  = 3 + 9 = 12                                   (Divisible By 3)

Step 2: Divisibility Test By 7

     The unit place of this number is 4, so this is an even number.

     Doubling 4, we get 4 × 2 = 8

     Remaining Portion of number = 9899

     Subtract double number            9899  -  8  =  9891

     Repeat the process: The unit place of new result is 1

     Doubling 1, we get 2

     Subtract it from new remaining portion      989  -  2  =  987

     Repeat the process: The unit place of new result is 7

     Doubling 7, we get 14

     Subtract it from new remaining portion      98  -  14  =  84

     Repeat the process: The unit place of new result is 4

     Doubling 4, we get 8

     Subtract it from new remaining portion      8  -  4  =  0        (Divisible By 7)

     So the given number 98994 is  divisible by 7.

Step 3: Since the given number is divisible by 3 as well as 7, so the given number is divisible by 21.

Verification: Now divide the number by 21 using conventional division method, we get that the given number 98994 is divisible by 21 and quotient is 4714 ‬with remainder 0.

22. Divisible By 22: If a number is divisible by 2 as well as divisible by 11, then the number is completely divisible by 22.

So it is very clear that the number should follow both the rules of divisibility test by 2 and divisibility test by 11 simultaneously.

Note: If number is not divisible by any one of 2 and 11, then number will not be divisible by 22. If number is divisible by 2 by not divisible by 11, it is not divisible by 22. Similarly, if number is not divisible by 2 but divisible by 11, it is not divisible by 22.

If difference of the sum of odd digits and the sum of even digits from a given number is divisible by 11, then the given number is divisible by 11.

First we will understand alternate places or even and odd places in a given number

(Figure: Alternative Digits in a Number)

Step for Test: Divisibility Test By 2: 1.      If last digit of number is 0,2,4,6,8 or number is even then it is divisible by 2. Then we proceed for next step      If last digit of number is neither of 0,2,4,6,8 or number is odd, then it is not divisible by 2. Then stop testing and declare that number is not divisible by 22. Divisibility Test By 11 2. Sum all odd alternate digits from the given number. Say it Odd Alternative Sum  3. sum all even alternate digits from the given number. Say it Even Alternative Sum 4. Subtract Even Alternative Sum from Odd Alternative Sum. Say it subtracted result. 5. If Subtracted Result is divisible by 11, then the given number is divisible by 11. If Number is divisible by 2 as well as 11, then number is divisible by 22.

For Example:

Let us check whether 9182690 is divisible by 22.

Divisibility Test By 2:

Since last digit of the number is '0', hence number is divisible by 2.   (Divisible By 2)

Divisibility Test By 11.

First sum alternative odd digits = 9 + 8 + 6 + 0 = 23

Then Sum alternative even digits  = 1 + 2 + 9 = 12

difference of two results = 23 -312 = 11

We can see that 11 is divisible by 11.                                                 (Divisible By 11)

So the given number 9182690 is divisible by 22.

Verification: Now divide the number by 22 using conventional division method, we get that the given number 9182690 is divisible by 22, and quotient is 417395 along ‬with remainder 0.

23. Divisible By 23: There are two rules for divisibility by 23. Whichever you get easy, you should apply in exams

Rule A: A number is divisible by 23 if result obtained from the addition of seven times the last digit (one's place) to the remaining part of the number is divisible by 23.

Rule B: A number is divisible by 23 if result obtained from the addition of three times the last two digits (combination of ten's place and one's place) to the remaining part of the number is divisible by 23.

Rule A: Steps for Test: 1. Write the last digit of number i.e. one's place of the number. 2. Multiply it by 7 3. Add this number to the remaining portion of the number. We will obtain a result. 4. Repeat above three steps for the result obtained many times until we get a number in two digits. 5. If final result is divisible by 23, then the given number is divisible by 23.  Rule B: Steps for Test: 1. Write the last two digits of number i.e. combination of ten's and one's place of the number. 2. Multiply it by 3 3. Add this number to the remaining portion of the number. We will obtain a result. 4. Repeat above three steps for the result obtained many times until we get a number less than 230. 5. If final result is divisible by 23, then the given number is divisible by 23.

For Example:

Let us check whether 93173 is divisible by 23.

Rule A: First Write last digit of the number i.e. 3

              Multiply it by 7 = 3 × 7 = 21

              Add this number in remaining portion = 9317 + 21 = 9338

              Repeat above steps

              Last digit of the new result = 8

              Multiply it by 7 = 8 × 7 = 56

              Add this number in remaining portion = 933 + 56 = 989

              Repeat above steps

              Last digit of the new result = 9

              Multiply it by 7 = 9 × 7 = 63

              Add this number in remaining portion = 98 + 63 = 161

              Repeat above steps

              Last digit of the new result = 1

              Multiply it by 7 = 1 × 7 = 7

              Add this number in remaining portion = 16 +7 = 23                   (Divisible By 23)

              Hence the given number 93173 is divisible by 23.

Rule B: First Write last two digits of the number i.e. 73

              Multiply it by 3 = 73 × 3 = 219

              Add this number in remaining portion = 931 + 219 = 1150

              Repeat above steps

              Write last two digits of the new number i.e. 50

              Multiply it by 3 = 50 × 3 = 150

              Add this number in remaining portion = 11 + 150 = 161            (Divisible By 23)

              Hence the given number 93173 is divisible by 23.

Verification: Now divide the number by 23 using conventional division method, we get that the given number 93173 is divisible by 23, and quotient is 4051 along ‬with remainder 0.

24. Divisible By 24: If a number is divisible by 8 as well as divisible by 3, then the number is completely divisible by 24.

So it is very clear that the number should follow both the rules of divisibility test by 8 and divisibility test by 3 simultaneously.

Note: If number is not divisible by any one of 8 and 3, then number will not be divisible by 24. If number is divisible by 8 by not divisible by 3, it is not divisible by 24. Similarly, if number is not divisible by 8 but divisible by 3, it is not divisible by 24.

Steps for Test: 1. Test for divisibility by 8.       If last digit of number is 0,2,4,6,8 or number is even then it is divisible by 2. Then we proceed for next step      If last digit of number is neither of 0,2,4,6,8 or number is odd, then it is not divisible by 2. Then stop testing and declare that number is not divisible by 24. 2. If number is divisible by 2, then write last three digits of the number. If this number is divisible by 8 then the given number is divisible by 8. Then we proceed to the next step.      If last three digit number is not divisible by 8, then stop testing. Number is not divisible by 8. 3. If Number is divisible by 8, then we proceed for step 3. 4. Test for Divisibility by 3      Sum all the digits      If sum found is more than one digit number, then again sum all the digits of the previous sum.     Repeat the process until we get sum of digits as a one digit number.     If final sum is divisible by 3, then the given number is divisible by 3.     If final sum is not divisible by 3, then the given number is not divisible by 3.  5. If number is divisible by 8 and 3, The given number is divisible by 24.

For Example:

Let us check whether 74558232 is divisible by 24.

Step 1: Divisibility Test By 2: 

Since last digit of number is 2 which is divisible by 2, so number is divisible by 2.

Step 2: Divisibility Test By 8: Write last three digits of the number = 232

since this number is divisible by 8, so number is divisible by 8.          (Divisible By 8)

Step 2: Divisibility Test By 3: 

Sum of All digits = 7 + 4 + 5 + 5 + 8 + 2 + 3 + 2 = 36

Again Sum of all digits = 3 + 6 = 9                                                       (Divisible by 3)

So the number is divisible by 3.

Since the given number is divisible by 8 and 3, Hence the given number 74558232 is divisible by 24.

Verification: Now divide the number by 24 using conventional division method, we get that the given number 74558232 is divisible by 24 and quotient is 3106593 ‬with remainder 0.

25. Divisible By 25: If combination of Last two place (tens place and unit place digits) forms a number divisible by 25, then whole number is divisible by 25.

Steps for Test: 1. First check whether last place of the number is 0,5 means number is divisible by 5. 2. If number is divisible by 5, then write last two digits (10's place and 1's place) as a number from the given number. 2. If this number is divisible by 25, then the given number is divisible by 25.     If this number is not divisible by 25, then the given number is not divisible by 25.

For Example: 7233550, 23456725, 975300, etc.

Example 1: 7233550

Last two digits = 50              (Divisible by 25)

So the complete number is divisible by 25.

Verification: Now divide the number by 25 using conventional division method, we get that the given number 7233550 is divisible by 25 and quotient is 289342 ‬with remainder 0.

26. Divisible By 26: If a number is divisible by 2 as well as 13, it is divisible by 26.

So it is very clear that the number should follow both the rules of divisibility test by 2 and divisibility test by 13 simultaneously.

Note: If number is not divisible by any one of 2 and 13, then number will not be divisible by 26. If number is divisible by 2 by not divisible by 13, it is not divisible by 26. Similarly, if number is not divisible by 2 but divisible by 13, it is not divisible by 26.

Steps for Test: 1. Check whether number is even or not, means its last digit is either 0, 2, 4, 6 or 8. If number is even then proceed to the next step. If number is not even then number is not divisible by 26 also. 2. If number is even then Write the last digit of number i.e. one's place of the number. 2. Multiply it by 4. 3. Add this number to the remaining portion of the number. We will obtain a result. 4. Repeat above three steps for the result obtained many times until we get a number less than 130. 5. If final result is divisible by 13, then the given number is divisible by 13.

For Example:

Let us check whether 262756 is divisible by 26.

Since unit place is 6, so number is even.                                    (Divisible by 2)

The unit place of this number is 3

Multiply it by 4 = 3 × 4 = 12

Remaining Portion of number =         26274

add the number multiplied by 4 =       26274  +   12  =  26286

Repeat the process: The unit place of new result is 6

Multiply it by 4 = 6 × 4 = 24

add this number in remaining portion =   2628  +   24  =  2652

Repeat the process: The unit place of new result is 2

Multiply it by 4 = 2 × 4 = 8

add this number in remaining portion =   265  +   8  =  273

Repeat the process: The unit place of new result is 3

Multiply it by 4 = 3 × 4 = 12

add this number in remaining portion =   27  +   12  =  39          (Divisible By 13)

Since the number is divisible by 2 as well as 13 both, So the given number 262756 is  divisible by 26.

Verification: Now divide the number by 26 using conventional division method, we get that the given number 262756 is  divisible by 26 and quotient is 10106 ‬with remainder 0.

27. Divisible By 27:  If sum of all digits in an integer is divisible by 27, then the number is divisible by 27.

Step for Test: 1. Sum all the digits 2. If sum found is more than one digit number, then again sum all the digits of the previous sum. 3. Repeat the process until we get sum of digits as a two digit number or you can stop up to three digit number less than 270. 4. If final sum is divisible by 27, then the given number is divisible by 27.     If final sum is not divisible by 27, then the given number is not divisible by 27.

For Example:

Let us check whether 162739989 is divisible by 27.

Sum of All digits = 1 + 6 + 2 + 7 + 3 + 9 + 9 + 8 + 9

                           = 54                      (Divisible by 27)

We can see that the sum of all digits in the given number is divisible by 27, so the given number 162739989 is divisible by 27.

Verification: Now divide the number by 27 using conventional division method, we get that the given number 162739989 is divisible by 27 and quotient is 6027407 along with remainder 0.

28. Divisible By 28: If a number is divisible by 4 as well as divisible by 7, then the number is completely divisible by 28. 

So it is very clear that the number should follow both the rules of divisibility test by 4 and divisibility test by 7 simultaneously.

Note: If number is not divisible by any one of 4 and 7, then number will not be divisible by 28. If number is divisible by 4 by not divisible by 7, it is not divisible by 28. Similarly, if number is not divisible by 4 but divisible by 7, it is not divisible by 28.

Steps for Test: 1. Check whether last two digit of the given number is divisible by 4.    If Yes then the given number is divisible by 4 then we proceed for next step.   If No then the given number is not divisible by 4 then not need to proceed further, the number is not divisible by 28 as well. 2. If Number is divisible by 4, then we proceed towards step 3. 3. Write the last digit of number i.e. one's place of the number. 4. Double it (Means multiply it by 2) 5. Subtract this number from the remaining portion of the number. We will obtain a result. 6. Repeat above three steps for the result obtained many times until we get a number less than 70. 7. If final result is divisible by 7, then the given number is divisible by 28.     If final result is not divisible by 7, then the given number is not divisible by 28.

For Example:

Let us check whether 98980 is divisible by 28.

The last two digits are 80 which is divisible by 4. so number is divisible by 4. (Divisible by 4)

First digit is 0.

Doubling 0, we get 0 × 2 = 0

Remaining Portion of number = 9898

Subtract double number            9898  -  0  =  9898

Repeat the process: The unit place of new result is 8

Doubling 8, we get 16

Subtract it from new remaining portion      989  -  16  =  973

Repeat the process: The unit place of new result is 3

Doubling 3, we get 6

Subtract it from new remaining portion      97  -  6  =  91

Repeat the process: The unit place of new result is 1

Doubling 1, we get 2

Subtract it from new remaining portion      9  -  2  =  7                        (Divisible By 7)

So the given number 98980 is  divisible by 7.

Since the given number is even as well as it is divisible by 7, so the given number 98980 is divisible by 28.

Verification: Now divide the number by 28 using conventional division method, we get that the given number 98980 is divisible by 28 and quotient is 3535 ‬with remainder 0.

29. Divisible By 29: A number is divisible by 29 if result obtained from the addition of three times the last digit (one's place) from the remaining part of the number is divisible by 29.

Steps for Test: 1. Write the last digit of number i.e. one's place of the number. 2. Multiply it by 3. 3. Add this number to the remaining portion of the number. We will obtain a result. 4. Repeat above three steps for the result obtained many times until we get a number less than 290. 5. If final result is divisible by 29, then the given number is divisible by 29.

For Example:

Let us check whether 61857 is divisible by 29.

The unit place of this number is 7

Multiply it by 3 = 7 × 3 = 21

Remaining Portion of number =         6185

add the number multiplied by 4 =       6185 + 21 = 6206

Repeat the process: The unit place of new result is 6

Multiply it by 3 = 6 × 3 = 18

add this number in remaining portion =   620 + 18 = 638

Repeat the process: The unit place of new result is 8

Multiply it by 3 = 8 × 3 = 24

add this number in remaining portion =   63  +   24  =  87              (Divisible By 29)

So the given number 61857 is  divisible by 29.

Verification: Now divide the number by 29 using conventional division method, we get that the given number 61857 is  divisible by 29 and quotient is 2133 ‬with remainder 0.

30. Divisible By 30: If number is divisible by 10 as well as 3, then number is divisible by 30.

Step for Test: 1. If one's place of number is '0', then number is divisible by 10 and then we will proceed to next step. If not, then number is not divisible by 30. 2. If number is divisible by 10, then proceed at step 3 for testing of divisibility by 3. 3. Sum all the digits 4. If sum found is more than one digit number, then again sum all the digits of the previous sum. 5. Repeat the process until we get sum of digits as a one digit number. 6. If final sum is divisible by 3, then the given number is divisible by 3.     If final sum is not divisible by 3, then the given number is not divisible by 3.  7. If number is divisible by 10 as well as 3, then number is divisible by 30.

For Example:

Example 1: Let us check whether 162738930 is divisible by 3.

Since one's place is 0, so it is divisible by 10.                         (Divisible By 10)

Sum of All digits = 1 + 6 + 2 + 7 + 3 + 8 + 9 + 3 + 0 =  

                           = 39

Again Sum all digits = 3 + 9 = 12

Again Sum all digits = 1 + 2 = 3         (Divisible by 3)

Since number is divisible by 10 as well as by 3, so the given number 162738930 is divisible by 30.

Verification: Now divide the number by 30 using conventional division method, we get that the given number 162738930 is divisible by 30 and quotient is 5424631‬ with remainder 0.